// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

// The computeRoots function included in this is based on materials
// covered by the following copyright and license:
//
// Geometric Tools, LLC
// Copyright (c) 1998-2010
// Distributed under the Boost Software License, Version 1.0.
//
// Permission is hereby granted, free of charge, to any person or organization
// obtaining a copy of the software and accompanying documentation covered by
// this license (the "Software") to use, reproduce, display, distribute,
// execute, and transmit the Software, and to prepare derivative works of the
// Software, and to permit third-parties to whom the Software is furnished to
// do so, all subject to the following:
//
// The copyright notices in the Software and this entire statement, including
// the above license grant, this restriction and the following disclaimer,
// must be included in all copies of the Software, in whole or in part, and
// all derivative works of the Software, unless such copies or derivative
// works are solely in the form of machine-executable object code generated by
// a source language processor.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
// SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
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// DEALINGS IN THE SOFTWARE.

#include <Eigen/Core>
#include <Eigen/Eigenvalues>
#include <Eigen/Geometry>
#include <bench/BenchTimer.h>
#include <iostream>

using namespace Eigen;
using namespace std;

template<typename Matrix, typename Roots>
inline void
computeRoots(const Matrix& m, Roots& roots)
{
	typedef typename Matrix::Scalar Scalar;
	const Scalar s_inv3 = 1.0 / 3.0;
	const Scalar s_sqrt3 = std::sqrt(Scalar(3.0));

	// The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0.  The
	// eigenvalues are the roots to this equation, all guaranteed to be
	// real-valued, because the matrix is symmetric.
	Scalar c0 = m(0, 0) * m(1, 1) * m(2, 2) + Scalar(2) * m(0, 1) * m(0, 2) * m(1, 2) - m(0, 0) * m(1, 2) * m(1, 2) -
				m(1, 1) * m(0, 2) * m(0, 2) - m(2, 2) * m(0, 1) * m(0, 1);
	Scalar c1 = m(0, 0) * m(1, 1) - m(0, 1) * m(0, 1) + m(0, 0) * m(2, 2) - m(0, 2) * m(0, 2) + m(1, 1) * m(2, 2) -
				m(1, 2) * m(1, 2);
	Scalar c2 = m(0, 0) + m(1, 1) + m(2, 2);

	// Construct the parameters used in classifying the roots of the equation
	// and in solving the equation for the roots in closed form.
	Scalar c2_over_3 = c2 * s_inv3;
	Scalar a_over_3 = (c1 - c2 * c2_over_3) * s_inv3;
	if (a_over_3 > Scalar(0))
		a_over_3 = Scalar(0);

	Scalar half_b = Scalar(0.5) * (c0 + c2_over_3 * (Scalar(2) * c2_over_3 * c2_over_3 - c1));

	Scalar q = half_b * half_b + a_over_3 * a_over_3 * a_over_3;
	if (q > Scalar(0))
		q = Scalar(0);

	// Compute the eigenvalues by solving for the roots of the polynomial.
	Scalar rho = std::sqrt(-a_over_3);
	Scalar theta = std::atan2(std::sqrt(-q), half_b) * s_inv3;
	Scalar cos_theta = std::cos(theta);
	Scalar sin_theta = std::sin(theta);
	roots(2) = c2_over_3 + Scalar(2) * rho * cos_theta;
	roots(0) = c2_over_3 - rho * (cos_theta + s_sqrt3 * sin_theta);
	roots(1) = c2_over_3 - rho * (cos_theta - s_sqrt3 * sin_theta);
}

template<typename Matrix, typename Vector>
void
eigen33(const Matrix& mat, Matrix& evecs, Vector& evals)
{
	typedef typename Matrix::Scalar Scalar;
	// Scale the matrix so its entries are in [-1,1].  The scaling is applied
	// only when at least one matrix entry has magnitude larger than 1.

	Scalar shift = mat.trace() / 3;
	Matrix scaledMat = mat;
	scaledMat.diagonal().array() -= shift;
	Scalar scale = scaledMat.cwiseAbs() /*.template triangularView<Lower>()*/.maxCoeff();
	scale = std::max(scale, Scalar(1));
	scaledMat /= scale;

	// Compute the eigenvalues
	//   scaledMat.setZero();
	computeRoots(scaledMat, evals);

	// compute the eigen vectors
	// **here we assume 3 different eigenvalues**

	// "optimized version" which appears to be slower with gcc!
	//     Vector base;
	//     Scalar alpha, beta;
	//     base <<   scaledMat(1,0) * scaledMat(2,1),
	//               scaledMat(1,0) * scaledMat(2,0),
	//              -scaledMat(1,0) * scaledMat(1,0);
	//     for(int k=0; k<2; ++k)
	//     {
	//       alpha = scaledMat(0,0) - evals(k);
	//       beta  = scaledMat(1,1) - evals(k);
	//       evecs.col(k) = (base + Vector(-beta*scaledMat(2,0), -alpha*scaledMat(2,1), alpha*beta)).normalized();
	//     }
	//     evecs.col(2) = evecs.col(0).cross(evecs.col(1)).normalized();

	//   // naive version
	//   Matrix tmp;
	//   tmp = scaledMat;
	//   tmp.diagonal().array() -= evals(0);
	//   evecs.col(0) = tmp.row(0).cross(tmp.row(1)).normalized();
	//
	//   tmp = scaledMat;
	//   tmp.diagonal().array() -= evals(1);
	//   evecs.col(1) = tmp.row(0).cross(tmp.row(1)).normalized();
	//
	//   tmp = scaledMat;
	//   tmp.diagonal().array() -= evals(2);
	//   evecs.col(2) = tmp.row(0).cross(tmp.row(1)).normalized();

	// a more stable version:
	if ((evals(2) - evals(0)) <= Eigen::NumTraits<Scalar>::epsilon()) {
		evecs.setIdentity();
	} else {
		Matrix tmp;
		tmp = scaledMat;
		tmp.diagonal().array() -= evals(2);
		evecs.col(2) = tmp.row(0).cross(tmp.row(1)).normalized();

		tmp = scaledMat;
		tmp.diagonal().array() -= evals(1);
		evecs.col(1) = tmp.row(0).cross(tmp.row(1));
		Scalar n1 = evecs.col(1).norm();
		if (n1 <= Eigen::NumTraits<Scalar>::epsilon())
			evecs.col(1) = evecs.col(2).unitOrthogonal();
		else
			evecs.col(1) /= n1;

		// make sure that evecs[1] is orthogonal to evecs[2]
		evecs.col(1) = evecs.col(2).cross(evecs.col(1).cross(evecs.col(2))).normalized();
		evecs.col(0) = evecs.col(2).cross(evecs.col(1));
	}

	// Rescale back to the original size.
	evals *= scale;
	evals.array() += shift;
}

int
main()
{
	BenchTimer t;
	int tries = 10;
	int rep = 400000;
	typedef Matrix3d Mat;
	typedef Vector3d Vec;
	Mat A = Mat::Random(3, 3);
	A = A.adjoint() * A;
	//   Mat Q = A.householderQr().householderQ();
	//   A = Q * Vec(2.2424567,2.2424566,7.454353).asDiagonal() * Q.transpose();

	SelfAdjointEigenSolver<Mat> eig(A);
	BENCH(t, tries, rep, eig.compute(A));
	std::cout << "Eigen iterative:  " << t.best() << "s\n";

	BENCH(t, tries, rep, eig.computeDirect(A));
	std::cout << "Eigen direct   :  " << t.best() << "s\n";

	Mat evecs;
	Vec evals;
	BENCH(t, tries, rep, eigen33(A, evecs, evals));
	std::cout << "Direct: " << t.best() << "s\n\n";

	//   std::cerr << "Eigenvalue/eigenvector diffs:\n";
	//   std::cerr << (evals - eig.eigenvalues()).transpose() << "\n";
	//   for(int k=0;k<3;++k)
	//     if(evecs.col(k).dot(eig.eigenvectors().col(k))<0)
	//       evecs.col(k) = -evecs.col(k);
	//   std::cerr << evecs - eig.eigenvectors() << "\n\n";
}
